Residual dip

Let us suppose p₀ is a dip picked at a point through the constrained non-linear optimization and p is the true dip at that point. The difference between them is a small residual dip $\Delta p$. 

$$p = p_0+\Delta p$$ (9)

We can substitute p in equation (2) with equation (9) and linearize the expression on the left-hand side of equation (2):  

$$\begin{array} {lll} P_{ij} & = & P(t+i\Delta t +jp \Delta x,... ...tyle{\partial P \over \partial t}j\Delta p \Delta x,\end{array}$$ (10)

to obtain P_ij as a linear function of the residual dip $\Delta p$.